The 1/Nc expansion is one of the very few methods we have for generating a systematic expansion of QCD at the energy scale relevant to hadron structure. The present formulation of this theory relies on the double-line notation for calculating the leading order of a diagram in the 1/Nc expansion, where the local SU(Nc) gauge symmetry is substituted by a U(Nc) symmetry and the associated U(1) ghost field is ignored. In the current work we demonstrate the insufficiency of this formulation for describing certain non-planar diagrams of interest in current attempts to model QCD. We derive a more complete set of Feynman rules that include the U(1) ghost field and provide a useful tool for calculating both color factors and 1/Nc orders of all color-singlet diagrams.